Minimum-energy Problem Research Articles

A relativistically covariant formulation in the theory of plasma stability

The definition of the ‘combined field’ and the ‘combined field potentials’ allows one to formulate a constrained minimum energy problem. The constraints can be put in a relativistically covariant form. The equations of stable equilibrium, for a non-rotating charged fluid, are obtained. If the physical conditions are suitably chosen, a special kind of Alfvén-like velocity may exist.

Minimum energy problem for discrete linear admissible control systems

The minimum energy problem for discrete linear admissible control system is solved as a quadratic programming problem for bounded variables. In this method, the optimal solution for the minimum energy problem of the non-admissible system is taken as the initial solution for the present problem. Minimum computer storage is required and no conditions are imposed on the matrix of the constraints. Numerical results indicate that the algorithm is a fast one.

Control Law for an Intercept System

A feedback control law is derived for a two-dimensional intercept system that combines in a step-by-step manner the closed-form solutions to a least-squares estimation of the target line speed and heading with a minimum control energy problem subject to a terminal intercept constraint. The major assumptions are that the pursuer possesses thrust modulation capabilities in addition to thrust vectoring, and that continuous measurements of the range coordinates are available for estimation purposes. Simulation results are included that indicate that good terminal accuracy is achieved when the estimation errors are small and illustrate the limitations on accuracy when fluctuations occur in the target speed and turn rate.

Stability of stationary rotation of a heavy solid body with two elastic rods: PMM vol. 40, n≗1, 1976, pp. 55–64

The example of the stability problem for stationary vertical rotation of a heavy “tuning fork” with a single point is used for showing that the method of stability investigation of a mechanical system stationary motions, based on the solution of the problem of minimum transformed potential energy of a system [1–6], is equally suitable for analyzing “trivial” and “nontrivial” stationary motions for which deformable elements of the system are, respectively, in the undeformed and the deformed state. Two methods for solving this problem are presented. Sufficient stability conditions that impose certain restrictions from below on the stiffness of rods and also from above and below on the angular velocity of the system uniform rotation are obtained and analyzed. The general statement of the problem of motion and stability of an elastic body with a cavity containing a fluid was presented by Rumiantsev [1]. The theorem on stability proved here is an extension of the Routh theorem to systems with distributed parameters, and reduces the problem of stability of stationary motion to that of minimum (transformed) potential energy W of the system. Solutions of a number of problems on the stability of stationary motions of a solid body with elastic rods and fluid in potential force fields appear in [2–5]. Two methods for establishing conditions for positive definiteness of the second variation δ 2 W in investigations of motion stability of mechanical systems consisting of absolutely rigid bodies and material points with attached deformable elastic and fluid bodies are presented and illustrated in [6] on the example of solutions of specific mechanical problems. The problem of stability of stationary motions of mechanical systems with distributed parameters were investigated in [7–10] (see the bibliography in [7–10]). The problem of stability of the “nontrivial” state of relative equilibrium of an absolutely rigid body with elastic rods is considered in [9], where the “trivial” and “nontrivial” equilibrium states of the system are characterized by the undeformed and deformed states, respectively. Only trivial states of relative equilibrium of rigid bodies with elastic rods were investigated in [2–6] and [7, 8]. It is noted in [9] that the previously used method based on the direct Liapunov method cannot be applied for solving problems of stability of the nontrivial state of relative equilibrium of the system, and another method of solution, based on the expansion of elastic displacements of rods in series of some complete system of functions is proposed. A finite number of first terms was retained in such series without any mathematical substantiation, and the system with distributed parameters is essentially reduced to some system with a finite number of degrees of freedom.

Suboptimal control of fixed-end-point minimum energy problem via singular perturbation theory

Suboptimal control of fixed-end-point minimum energy problem via singular perturbation theory

Optimisation of mine winder: state-constrained minimum-energy problem

In the application of modern control theory to mining processes, the optimal control of winding engines has recently been investigated. This letter sets the problem in a general context, and derives the optimal minimum-energy controller when the system is subject to hard state constraints on the midshaft speed.

Use of piecewise linearization for suboptimal control of non-linear systems†

This paper develops the theory for solving piecewise-linear optimal control problems and then employs this theory to obtain suboptimal solutions for controlling non-linear systems. The non-linear system is first approximated by a piecewise-linear model. A jump condition on the adjoint variables and a continuity condition on the Hamiltonian are derived and utilized to obtain the optimal solution to the piecewise-linear model. This control is then considered as a suboptimal solution to the non-linear problem. The piecewise-linear modelling technique provides a means of decoupling the system and adjoint equations and hence provides a simple means of calculating the suboptimal control. A variable mass, minimum-energy problem is solved to illustrate the technique. A feedback scheme is introduced to minimize the boundary and cost errors.

An Historical Survey of Computational Methods in Optimal Control

The last twenty years have seen a great flourishing in optimal control theory. In this paper, we shall highlight some of the salient theoretical developments in the specific area of algorithms. The first algorithms for optimal control were aimed at unconstrained problems and were derived by using first and second variation methods of the calculus of variations. These methods have subsequently been recognized as gradient, Newton–Raphson, or Gauss–Newton methods in function space. A much more recent addition to the arsenal of unconstrained optimal control algorithms are several variations of conjugate gradient methods. At first, constrained optimal control problems could only be solved by exterior penalty function methods. Later algorithms specifically designed for constrained problems have appeared. Among these we find methods for solving the unconstrained linear quadratic regulator problem as well as certain constrained minimum time and minimum energy problems. Differential-dynamic programming was developed from dynamic programming considerations. The conditional gradient method, the gradient projection method and a couple of feasible directions methods were obtained as extensions or adaptations of related algorithms for finite-dimensional problems. Finally, the so-called $\varepsilon $-methods combine the Ritz method with penalty function techniques. Some of the recent work has dealt with discretization effects. This has led to the concept of well-posedness of a problem and to adaptive integration techniques.

The thermodynamic approach to fluidized drying and moistening optimization

AbstractFor drying and moistening processes in batch fluidization and fluidization in a horizontal exchanger, the overall mathematical model describing simultaneous heat and mass exchange is given. On this basis, the problem of finding the optimal temperatures of inlet gas is formulated and investigated for the case when inlet gas humidity is constant. It is explained that when unit economical values of gas and solid phases are linear with regard to unit available energies (exergies), the economical problem of minimum costs is equivalent to the thermodynamical problem of minimum available energy (exergy) dissipation. Therefore, the performance index expression in exergy terms is formulated as a functional of the Bolza form. The dynamic programming method is used to obtain results of optimal trajectories and decisions. The dependence of optimal process time on apparatus price, which is characterized here by a Lagrangian multiplier λ, is pointed out.The results indicate that the constant gas inlet temperature policy commonly applied to the processes considered here prevents attainment of the process optimum. In fact, increasing inlet temperature until an admissible gas temperature is attained is the optimal policy.

Sensitivity approaches to optimization of linear systems with time delay

This paper presents two different sensitivity approaches to synthesis problems of optimal control for linear systems with small time delays. In the first part, the optimal control considering sensitivity is discussed for a minimum energy problem to make up for the undesirable effect caused by a small time delay. In this sensitivity synthesis of optimal control, the central role is played by a combined system which consists of the model neglecting the delay time and the sensitivity model with respect to it. In order to ensure the optimal control with zero-sensitive terminal constraints, the controllability property of the combined system is investigated, and the necessary and sufficient condition for it is derived for a linear system. The second part presents a synthesis method of a suboptimal control for a regulator problem, in which the optimal control is expanded into MacLaurin series in terms of the delay time and the first two or more terms are used to yield a sub-optimal control. Another slightly different approach, in which the singular perturbation method is applied, is also discussed and compared with the other one. As shown by the numerical examples, the present sensitivity approaches decrease considerably the computational efforts, while preserving a good performance in the synthesis of time delay systems.

Liquid bridges between cylinders, in a torus, and between spheres

AbstractThe stability of a capillary liquid bridge of given volume between two small, solid, equal, separated spheres is investigated by formulating and treating a minimum energy problem in the calculus of variations and by experiment. A conjecture is made that in the case of two solutions one and only one is minimizing, and that the case of one solution represents the limiting stable bridge. This theory agrees accurately with our stability experiments. Furthermore, it is possible to predict the cohesive force. For the case of spheres in contact, the theory presented here is in agreement with some experimental work and also with the theory of Fisher and calculations of cohesive force based on Melrose and Wallick's solution to the bridge problem. For the case of separated spheres, the agreement with the only available experimental data is excellent except for close separations.

A conforming triangular finite element plate bending solution

AbstractA conforming plate bending solution using simple polynomial deflection functions of third‐degree inside each triangular element is presented. In order to avoid normal slope discontinuities along the sides of the elements, the plate displacement parameters are subjected to ‘slope continuity conditions’ acting as constraints to the minimum potential energy problem. This is then solved by the classical method of Lagrange introducing multipliers as new auxiliary variables. If a special variational formulation of the problem is used, it can be shown that the Lagrangean multipliers are generalized stress parameters. The suggested solution is therefore basically a ‘mixed’ solution, the unknown variables of the problem being both displacement and stress parameters. Several numerical results are presented.

Analytical results on the use of reduced models in the control of linear dynamical systems

Two sets of first-order differential equations denoted by X and Xr are assumed to exist, where X and Xr govern the behaviour of the plant and the reduced model, respectively. The performance of the plant when it is controlled on the false assumption that its behaviour is governed by Xr is investigated. There optimal-control problems are considered:(a) fixed-target fixed-terminal-point minimum-energy problem(b) fixed-target free-terminal-point minimum-energy tracking problem(c) output-regulation problem.Explicit relations are obtained for the deterioration in performance in model control of the plant. In the first two problems, the interconnection between the error functional commonly adopted for the model-reduction problem and the deterioration in performance is demonstrated; the results are applied to reduced models obtained by the method of optimal projection along invariant subspaces.

Minimum energy problems in Hilbert function space

Minimum energy problems in Hilbert function space are formulated. The methods presented are applicable to both continuous and discrete linear systems. Transformation of a particular operator equation into a set of 2n linear differential equations is described. Applications to physical systems are illustrated.

Some Aspects of the Minimum-Energy Problem in General Relativity

It has been shown that the energy of a charged point mass is positive-definite, and that its metric expressed in isotropic coordinates represents a minimum-energy state. In this paper it is found that for an electric field about this particle which can be derived from a scalar potential, this energy is stationary regardless of the form of gravitational wave interaction, in first approximation (small-perturbation theory). Thus, it is expected that for certain metrics, such a state has lower energy, in first order, than one in which any weak external gravitation is included.

On the existence of inflection points in minimum energy problems

Existence of inflection points in cost vs terminal time curve for linear minimum energy regulator

A new approach to a general minimum energy problem

A new approach to a general minimum energy problem